An Effective Model for the Flat Bands of a Dice Lattice

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Master Thesis

An Effective Model for the Flat Bands of a Dice Lattice

Lena Engstr¨ om

Condensed Matter Theory, Department of Theoretical Physics, School of Engineering Sciences

Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2017

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Akademisk avhandling f¨ or avl¨ aggande av teknologie masterexamen inom ¨ amnesomr˚ adet teoretisk fysik.

Scientific thesis for the degree of Master of Engineering in the subject area of The- oretical physics.

TRITA–FYS 2017:49 ISSN 0280-316X

ISRN KTH/FYS/–17:49SE Lena Engstr¨ c om August 2017

Printed in Sweden by Universitetsservice US AB, Stockholm August 2017

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Abstract

In lattice models with flat (dispersionless) Bloch bands ground state properties will be dominated by interactions between particles. Flat bands possess the property that particles will be localized to a few sites in the lattice. If interactions are added to such a system, particles will only interact with the few neighbouring states with which they overlap. Not only does the localization result in the possibility for exotic phases to emerge, but it also allows for the interactions to be treated through a projection onto the localized states. In this work the, largely unexplored, flat band limit of the dice lattice is introduced and an effective model for the low-energy states of a generic Hubbard-type Hamiltonian is derived. We then specialize to the case of attractively interacting fermions. In addition, we characterize the ground states in the few-particle limit. The effective model includes several interaction terms, on-site and nearest neighbour terms as well as interactions on triangles.

Fermion pairs will dominate for an attractive system. The model is mapped onto a ferromagnetic Heisenberg Hamiltonian which is valid for states containing only pairs. Further, a prediction for many-body ground states at any filling of the lattice is made through a perturbative approach. The full ground state for few particles is found to be dominated by pairs uniformly distributed in the lattice. In summary, we here present a simple spin model that is shown to describe the dominating properties arising from an attractive interaction in a flat band.

Key words: flat bands, Dice lattice, 2D lattices, Ultracold gases, Hubbard model, Heisenberg model, effective Hamiltonian, projection

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I gittermodeller med platta (dispersionsl¨ osa) Bloch band kommer grundtillst˚ andets egenskaper att vara dominerade av interaktioner mellan partiklar. Platta band har egenskapen att partiklar kommer att vara lokaliserade till ett f˚ atal punkter i gittret.

Om interaktioner l¨ aggs till i ett s˚ adant system, interagerar partiklar bara med de f˚ a angr¨ ansade tillst˚ anden som de ¨ overlappar med. Lokalisering resulterar inte bara i m¨ ojligheten f¨ or exotiska faser att uppst˚ a utan det till˚ ater ¨ aven interaktioner att be- handlas via en projektion till de lokaliserade tillst˚ anden. Detta arbete introducerar det, till st¨ orsta dels outforskade, t¨ arningsformade gittret och h¨ arleder en effektiv modell f¨ or l˚ agenergitillst˚ aden f¨ or en generisk Hubbardmodell. Vi studerar sedan s¨ arskilt fallet med attraktivt interagerande fermioner. Dessutom karakt¨ ariserar vi grundtillst˚ anden i gr¨ ansen med f˚ a partiklar. Den effektiva modellen inkluderar flera interaktionstermer, interaktioner p˚ a samma punkt i gittret, interaktioner mellan angr¨ ansande punkter, och interaktioner i triangelformer. Fermioner som bildar par kommer att dominera i det attraktiva systemet. Modellen avbildas till en ferromag- netisk Heisenberg Hamiltonian, vilken kommer att vara giltig i system som bara inneh˚ aller par. Vidare g¨ ors en f¨ oruts¨ agelse om m˚ angpartikeltillst˚ anden f¨ or godtyck- lig fyllning av gittret genom ett perturbativt tillv¨ agag˚ angss¨ att. Det fullst¨ andiga grundtillst˚ andet f¨ or f˚ a partiklar domineras av par j¨ amnt distribuerade i gittret.

Sammanfattningsvis s˚ a presenteras h¨ ar en enkel spinmodell som visas beskriva de dominerade egenskaperna som uppkommer fr˚ an en attraktiv interaktion i ett platt band.

Nyckelord: platta band, t¨ arningsformat gitter, 2D gitter, ultrakalla gaser, Hub- bard modellen, Heisenberg modellen, effektiv Hamiltonian, projektion

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Acknowledgements

I would like to express my gratitude to my supervisor Professor Sebastian Huber at ETH Z¨ urich for giving me the opportunity to do this project. With his valuable insights and experience in the field his guidance steered the work into its best possible version. I owe him many thanks for the introduction into this exciting field of physics.

I would also like to thank Murad Tovmasyan for his guidance and help, without which I would have been quite lost. He has invested his time and full effort throughout this project, by making himself available for discussions and providing a sound theoretical background.

I also extend a thank you to my examiner at KTH, Professor Jack Lidmar, for all the information provided and the help with the formalities concerning the thesis.

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Introduction

Lattice models with completely flat Bloch bands offer systems where interacting ground state properties can be studied in detail. In a flat bands all particles will have a vanishing group velocity v =

^∂E_∂k

= 0. That is to say that the eigenstates are immobile and do not contribute to the transport properties of the system. Particles are then localized to a limited number of sites in the lattice. This localization is however a single particle effect. Interactions between particles will instead dominate the properties and transport becomes possible through the interaction effects.

To characterize these properties the goal is to study how particles interact on such flat bands. Interacting particles do not have well-defined separable single particle states since a particle affects the states of the other particles. The interaction effects can for the flat bands be characterized with methods otherwise unavailable.

Since the non-interacting states are localized in the flat band an effective model can be given for low energies. This thesis makes predictions for the low-energy states of the dice lattice by projecting interactions onto the lowest flat band. The model and lattice that describe the dominant processes can then be greatly simplified.

The dice lattice, also called the T

3

lattice, is in this thesis studied for a tight binding model. The non-interacting model consists of three bands which under a magnetic field, with a specific value, can become separated and flat. The flatness of the bands arises from frustration in the hopping elements. Frustrated hopping is an interference effect where classical paths for the particles interfere destructively due to the geometric frustration. Eigenstates in this model will then be restricted to a limited number of sites. Other bipartite lattices of the same class, such as the Kagome lattice, the Lieb lattice, and the Creutz ladder, similarly result in frustrated hopping [1–6]. Generally in hopping problems the low energy physics is well described by the long-wavelength part of the dispersion relation. However, due to the flatness of the bands the states will behave quite differently here.

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The effect of an on-site Hubbard interaction U is studied by projecting the interaction onto the localized eigenstates of the non-interacting model. The projection results in an low-energy effective model with a larger number of interaction terms.

The interaction constant U can be chosen to be either repulsive U > 0 or attractive U < 0. The model for hard-core bosons with a repulsive interaction has been studied for the dice lattice [7–10] and results in a supersolid at low filling. The fermionic model is here studied further. For an attractive interaction the formation of pairs will be beneficial. However, since there are terms that break pairs the effect from these terms on the full state must be evaluated.

The two following chapters derive the effective model starting with a simple tight binding model. In chapter 2 the properties of the dice lattice is calculated, by find- ing the band structure and the localized eigenstates. In chapter 3 the interaction is projected for both bosons and fermions. In the two next chapters the full effective model is evaluated to find the low-energy ground state for low fillings. In chapter 4 the pair-breaking terms are removed. Under the assumption that the model only contains pairs the Hamiltonian can be mapped onto a spin-1/2 Heisenberg model.

In chapter 5 the ground state of the Hamiltonian for low filling is evaluated by

studying the two-particle and two-pair problem by splitting the Hamiltonian in

two parts. The ground state of the non-perturbed Hamiltonian is found and the

possible corrections from second order perturbation theory are approximated. As

a conclusion, we will see that due to the small size of the perturbation the simple

spin model can be considered to describe the main contributions to the many-body

ground state at arbitrary filling.

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Chapter 2

Dice Lattice

In this chapter the non-interacting tight binding model for the dice lattice will be solved analytically. Both the bosonic and fermionic case will be considered. All three bands of the model will be seen to become flat and separated when a specific magnetic field is applied. Firstly, the energy bands for the lattice without any applied magnetic field are calculated. Secondly, hopping amplitudes are engineered in such a way that half a magnetic flux unit passes through each rhombus in the lattice for a chosen gauge. This gauge will be chosen so that all hopping amplitudes have a real value. The eigenstates will be given as Bloch states that are transformed into Wannier functions in real space. Finally, these Wannier functions are seen to describe eigenstates that are localized to a few sites in the lattice, i.e. have compact support.

2.1 The tight-binding model

The tight binding model describe single particles that can occupy sites in the lattice and that can hop to a neighbouring site with a hopping amplitude t [11]. These amplitudes can vary depending on what bond a particle hops over and is more generally given as t

i,j

for the bond i, j. For bosons the tight binding Hamiltonian with nearest neighbour hopping is:

H = − ˆ X

hi,ji

t

_i,j

ˆ b

^†_i

ˆ b

_j

+ t

^∗_i,j

ˆ b

^†_j

ˆ b

_i

= − X

hi,ji

t

_i,j

ˆ b

^†_i

ˆ b

_j

+ h.c.

, (2.1)

where the annihilation operator ˆ b

j

annihilates a particle at site j and the creation operator ˆ b

^†_i

creates a particle at site i. The bosonic operators are defined by the

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commutation relations:

hˆb

i

, ˆ b

^†_j

i

= ˆ b

i

ˆ b

^†_j

− ˆb

^†_j

ˆ b

i

= δ

i,j

, hˆb

i

, ˆ b

j

i

= 0, hˆb

^†_i

, ˆ b

^†_j

i

= 0, (2.2) where:

δ

_i,j

( 1, i = j

0, i 6= j . (2.3)

In this number basis, where the states are described by the particle occupation n

i

at each site, the states are described as operators acting on a vacuum state to create a configuration of particles. If the hopping amplitude t is real and has the same value along any bond in the lattice the Hamiltonian can be simplified to:

H = −t ˆ X

hi,ji

ˆb

^†_i

ˆ b

j

+ h.c.

. (2.4)

For fermions the Hamiltonian includes fermionic operators with an additional spin- component σ. The fermionic operators are defined from the anticommutation relations:

n ˆ c

i,σ

, ˆ c

^†_j,σ0

o

= ˆ c

i,σ

c ˆ

^†_j,σ0

+ ˆ c

^†_j,σ0

ˆ c

i,σ

= δ

i,j

δ

σ,σ⁰

, {ˆ c

i,σ

, ˆ c

j,σ⁰

} = 0, n ˆ c

^†_i,σ

, ˆ c

^†_j,σ0

o

= 0.

(2.5) The fermionic Hamiltonian then becomes:

H = − ˆ X

hi,ji

t

^σ_i,j

ˆ c

^†_i,σ

c ˆ

_j,σ

+ h.c.

. (2.6)

To preserve time-reversal symmetry it is required that the hopping amplitudes for opposite spins are each others complex conjugates t

^σ_i,j⁰

= t

^σ_i,j

∗

, where σ = −σ

⁰

. The resulting band structure of the model will depend on the geometric structure of the lattice. The number of bands is given by how many sites are included in a unit cell for these to form a Bravais lattice. The shape of the bands and the gap separating them will depend on the unit vectors for the lattice. In each unit cell j there will be one site which belongs to the sublattice m. There are then operators for each sublattice in each unit cell ˆ c

j,m,σ

. The hopping occurs both between sublattices and between unit cells. Without any further interaction this model can be solved by a Fourier transform of the Hamiltonian since the hopping will be diagonal in the reciprocal k-space. The Fourier transform of the operators takes the form:

ˆ

c

^†_j,m,σ

= 1

√ N

X

k∈B.Z.

e

^−ik·j

c ˆ

^†_k,m,σ

, ˆ c

j,m,σ

= 1

√ N

X

k∈B.Z.

e

^ik·j

c ˆ

k,m,σ

. (2.7)

From here on the fermionic operators will be used for all non-interacting states. The

corresponding bosonic operators will differ to the fermionic by the commutation re-

lations, where the bosonic commute and the fermionic anticommute, according to

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2.2. Band structure without magnetic field 5

Figure 2.1: The dice lattice. There are two different types of sites, sixfold coordinated (circles) and threefold coordinated (triangles).

equations (2.2) and (2.5). An arbitrary number of bosons can occupy a site in the lattice, while the Pauli principle restricts the fermions to only allowing occupation of one up-spin and one down-spin fermion at each site. From the tight-binding Hamiltonian the eigenstates ψ will be given by the Schr¨ odinger equation ˆ Hψ = Eψ.

In this thesis the dice lattice is considered. The dice lattice is a two dimensional lattice which has three sites in the unit cell and therefore three bands in the non- interacting tight binding model. However, once a magnetic field is added the band structure will change as the field modifies the structure of hopping amplitudes in the lattice.

2.2 Band structure without magnetic field

The dice lattice can be seen in figure 2.1. The lattice constant a is given by the

length of the link between two neighbouring sites. In the unit cell of the dice

lattice three sites are included: one with sixfold coordination (∗) and two with

threefold coordination (∆). The unit vectors connect the unit cell from the sixfold

coordinated site to another of the same type in the neighbouring unit cells, and are

given as:

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(η

₁

= √ 3a(1, 0) η

2

= √

3a

1 2

,

√3 2

. (2.8)

In figure 2.2 the unit cell is marked out along with its unit vectors. The unit cells are located on an underlying triangular Bravais lattice.

The band structure of a simple hopping Hamiltonian in the dice lattice is given by the Schr¨ odinger equation for the three sites in the unit cell. The eigenvalues of the Hamiltonian are then given by the three bands:

(

0

(k) = 0

±

(k) = ± √

2p3 + 2 cos (k

1

) + 2 cos (k

2

) + 2 cos (k

3

), (2.9) where



 

 

k

1

= k · η

1

k

₂

= k · η

₂

k

3

= k · (η

2

− η

1

)

(2.10)

are the reciprocal unit vectors and a vector that combines these.

The dice lattice is a bipartite lattice, meaning that the sites can be divided into two disjoint sublattices. For this lattice these are the sixfold and threefold connected sites. The sixfold connected sites only have bonds to threefold connected sites and the other way around. Completely flat bands are achieved by adding a magnetic field perpendicular to the lattice at the right value to form Aharonov-Bohm cages that the particles are localized in [12]. Compared to the = 0 band, that is flat for the lattice without any magnetic field, there is no band touching when all bands are flat. For the separated flat bands it is possible to study localized states in only that band. For bipartite lattices the flat bands contain well localized states [13].

2.3 Aharonov-Bohm Cages

If an external magnetic field is applied the Hamiltonian will be affected by the vector potential A, connected to the magnetic field as B = ∇ × A:

H = i~

2m

^∗

∇ + V → 1

2m

^∗

(i~∇ − eA)

²

+ V. (2.11) On the form of a tight binding Hamiltonian this means that the hopping amplitude t is affected by the field [12]. Peierls’ substitution is derived from the shifted Hamiltonian. The substitution adds a phase to the hopping amplitude:

t → te

⁻ⁱ^~^e^{R A·dr}

= te

^−i2πΦ^B^/Φ⁰

. (2.12)

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2.3. Aharonov-Bohm Cages 7

 η

1

 η

2

Figure 2.2: Unit cell of the dice lattice with unit vectors.

Where Φ

0

= ~/e is the unit flux. This means that a particle picks up a phase as it hops over a bond. For a homogeneous magnetic field the flux over a plaquette is set to a fixed quantity and is given by [14]:

Φ = I

cell

A · dr = 2πν. (2.13)

To get a system of flat bands the value ν = 1/2 is chosen, so that a flux of Φ = π passes through each plaquette [15, 16]. The hopping amplitudes of the lattices are then to be modified so that a particle picks up a phase π from hopping along the bonds surrounding one elementary rhombus of the dice lattice.

The physical system is gauge invariant since the vector potential results in the same magnetic field B even after a gauge transformation A → A + ∇f . The phases can then be chosen in any way as long as π-flux passes through each plaquette. A real gauge can be chosen so that all Aharonov-Bohm phases result in real hopping amplitudes. The ν = 1/2 flux can be achieved in the dice lattice by adding the phase ±π for one bond on each elementary rhombus. The hopping amplitude on that bond then shifts from t → e

^±iπ

t = −t. The chosen gauge is shown in figure 2.3 where the bonds with negative hopping amplitude are denoted by a line added on that bond.

Since all the hopping amplitudes have a real value, time reversal symmetry will

be preserved by taking t

^σ⁰

= [t

^σ

]

^∗

= t, the same amplitude for all spins. With these

modified hopping amplitudes a new unit cell must be chosen, since the translational

symmetry of the lattice has been broken.

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2.3.1 Magnetic Translational Operator

For the zero-field case the lattice translation operators ˆ T

_i⁰

commute with the Hamil- tonian and with each other as h ˆ T

₁⁰

, ˆ T

₂⁰

i

= 0. The operators are defined as:

T ˆ

₁⁰

= X

m,n

ˆ b

^†_m+1,n

ˆ b

_m,n

T ˆ

₂⁰

= X

m,n

ˆ b

^†_m,n+1

ˆ b

_m,n

. (2.14)

However, when an external magnetic field is added the translational symmetry of the lattice is broken and the operators are shifted into new magnetic translation operators [14]:

T ˆ

₁^M

= X

m,n

ˆ b

^†_m+1,n

ˆ b

m,n

e

^iθ¹^m,n

T ˆ

₂^M

= X

m,n

ˆ b

^†_m,n+1

ˆ b

m,n

e

^iθ²^m,n

. (2.15)

The requirements for these are that they should commute with the Hamiltonian, that has modified hopping amplitudes, and with each other. The operators should then be constructed so that they enclose a super-cell of dimension k × l on which an integer multiple of the flux 2π is enclosed. To fulfil the requirements of the operators the condition for this super-cell is:

e

^−iklΦ

ˆ T

₁^M

^k

ˆ T

₂^M

^l

= ˆ T

₁^M

^l

ˆ T

₂^M

^k

. (2.16)

The smallest possible super-cell is the magnetic unit cell, which is given by kl = 1/ν, where ν is given by equation (2.13). For the case when ν = 1/2 this means that the magnetic unit cell has twice as large area as the unit cell for the zero-field case.

The new unit cell that has been chosen can be seen in figure 2.3. The unit cell now includes six sites: two sixfold coordinated sites and four threefold coordinated sites. For this gauge choice and unit cell the C3 symmetry of the lattice has been broken, so the unit cells are now placed on an underlying rectangular lattice. The band structure will now be seen to be flat for this model.

2.3.2 Band structure

After adding a flux of π through each plaquette the unit cell will consist of twice as many sites. There are now six sites in each unit cell and six bands are expected to arise from the hopping Hamiltonian. As will be seen these are three doubly degenerate energy levels. For this specific value of the flux all of these bands will be flat. The Hamiltonian matrix is given from writing the Schr¨ odinger equation on matrix form:

H(k)ψ = Eψ, (2.17)

where the solution is then a vector of the operators for each site in the unit cell

ψ = α · [ˆ c

_k,1

, ˆ c

_k,2

, ˆ c

_k,3

, ˆ c

_k,4

, ˆ c

_k,5

, ˆ c

_k,6

]. α is an eigenvector to the matrix H(k).

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2.3. Aharonov-Bohm Cages 9

1 2

3 4

5 6

Figure 2.3: Unit cell of dice lattice with π-flux passing through each rhombus. The

unit cell contains 6 sites which are numbered.

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The eigenvalues and eigenstates of the tight binding Hamiltonian are given by the matrix for the hopping in and between the new unit cells (where t = −1):

H(k) =

=







0 1 − e

^−ik·e^x

1 + e

^−ik·e^x

0 1 e

^ik·e^y

1 − e

^ik·e^x

0 0 e

^ik·e^y

0 0

1 + e

^ik·e^x

0 0 −1 0 0

0 e

^−ik·e^y

−1 0 1 + e

^ik·e^x

−1 + e

^ik·e^x

1 0 0 1 + e

^−ik·e^x

0 0

e

^−ik·e^y

0 0 −1 + e

^−ik·e^x

0 0





 ,

(2.18) where

( e

x

= √ 3a(1, 0) e

_y

= √

3a 0, √

3 (2.19)

are the unit vectors for the new unit cells. These reach from a sites with a specific number to the corresponding sites in neighbouring unit cells. The Brilloiun zone constructed from new reciprocal lattice vectors, is given from the condition:

K

i

· e

j

= 2πδ

ij

. (2.20)

The reciprocal unit vectors are:

( K

x

=

^√^2π

3a

(1, 0)

K

_y

=

^2π_3a

(0, 1). (2.21)

The first Brillouin zone is then given by: k

x

∈ [−

^√^π

3a

,

^√^π

3a

) and k

y

∈ [−

_3a^π

,

_3a^π

). The eigenvalues of the Hamiltonian matrix can be found by constructing the following vectors e

i

= (n

1

, . . . , n

i

, . . . , n

N

)

^T

where n

i

= 1 and n

j

= 0 for j 6= i. If the matrix H(k) now acts twice on the vectors e

1

and e

4

, it can be seen that these are eigenvectors to the H(k)

²

matrix:

H(k)

²

e

1

= 6e

1

H(k)

²

e

4

= 6e

4

. (2.22) Two doubly degenerate eigenvalues to the Hamiltonian are = ± √

6, as the eigenvalues of H(k)

²

should be given by

²

= 6. A third doubly degenerate eigenvalue is found for = 0. All diagonal elements of H(k) are 0 meaning that the sum of all eigenvalues must be 0. For the H(k)-matrix the following three doubly degenerate eigenvalues are then found:

(

0

(k) = 0

±

(k) = ± √

6. (2.23)

The model then has a band gap of ∆ = √

6, from each doubly degenerate band to

the next. Notably, ∆ is also the bandgap from the lowest band to the next. Models

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2.3. Aharonov-Bohm Cages 11 only considering the lowest energy states should be considered for energies below this energy gap as not to include intraband effects. For each degenerate band two eigenstates will be found.

2.3.3 Bloch states

The eigenvectors in the lowest band ( = − √

6) can be found by constructing them as:

|v

1

i = 1

√ 2

H(k)

√ 6 − I

|e

1

i, |v

2

i = 1

√ 2

H(k)

√ 6 − I

|e

4

i, (2.24) which written in operators is:

|v

1

i = h

−

^√¹

2

,

^1−e^√^ik·ex

12

,

^1+e^√^ik·ex

12

, 0,

^√¹

12

,

^e^−ik·ey^√

12

i ˆ c

k

|v

2

i = h 0,

^e^ik·ey^√

12

, −

^√¹

12

, −

^√¹

2

,

^1+e^√^−ik·ex

12

,

^−1+e^√^−ik·ex

12

i ˆ c

k

,

(2.25)

where ˆ c

k

= [ˆ c

k,1

, ˆ c

k,2

, ˆ c

k,3

, ˆ c

k,4

, ˆ c

k,5

, ˆ c

k,6

] is a vector of the operators for each site in a unit cell. There will be two eigenvectors for the each eigenvalue since the band is doubly degenerate.

Similarly to the lower band the eigenstates for the upper band ( = √

6) can be constructed as:

|u

1

i = 1

√ 2

H(k)

√ 6 + I

|e

1

i, |u

2

i = 1

√ 2

H(k)

√ 6 + I

|e

4

i, (2.26) which is:

|u

1

i = h

√1

2

,

^1−e^√^ik·ex

12

,

^1+e^√^ik·ex

12

, 0,

^√¹

12

,

^e^−ik·ey^√

12

i ˆ c

k

|u

2

i = h 0,

^e^√^ik·ey

12

, −

^√¹

12

,

^√¹

2

,

^1+e^√^−ik·ex

12

,

^−1+e^√^−ik·ex

12

i ˆ c

_k

. (2.27) The eigenstates to the zero band are not as straight forward to find as for the other bands. Consider the projectors of each band that projects any state onto the band:

P

₋

= |v

₁

ihv

₁

|+|v

₂

ihv

₂

|, P

₊

= |u

₁

ihu

₁

|+|u

₂

ihu

₂

|, P

₀

= |z

₁

ihz

₁

|+|z

₂

ihz

₂

|.

(2.28) All the bands should span the total Hilbert space of the system. The projectors into the subspace of each band should then have the completeness relation:

P

+

+ P

−

+ P

0

= I. (2.29)

This condition will be fulfilled for the following Bloch states:

|z

1

i = √

¹

18−12 cos (k·ex)

0, −2i sin (k · e

x

) − e

^ik·e^y

, 2 cos (k · e

x

) − 3, 0, 0, 1 − e

^−ik·e^x

+ e

^−ik·e^y

+ e

^−i(k·e^x^+k·e^y⁾

ˆ c

k

|z

2

i = √

¹

18−12 cos (k·e_x)

0, 1 − e

^ik·e^x

+ e

^ik·e^y

+ e

^i(k·e^x^+k·e^y⁾

, 0, 0, 2 cos (k · e

_x

) − 3, −2i sin (k · e

_x

) + e

^−ik·e^y

ˆ c

_k

.

(2.30)

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When these are multiplied by the matrix of the Hamiltonian the eigenvalue is 0.

2.3.4 Chern numbers for bands

The Chern number C is a topologically invariant quantity that is defined for an isolated band s. The topological invariance means that the system cannot adia- batically be transformed into a system with a different Chern number. The Chern number is for that reason often used as a topological index for the band. It has been proven that for the three properties in tight binding models: a flat band, a non-zero Chern number, and local hopping, all cannot be fulfilled at once [17]. The dice lattice model already has two of these properties, nearest neighbour hopping and flat bands. The Chern number is then expected to vanish. The Chern number is a property derived from the Berry phase. The Berry phase describes the phase that a state picks up in parameter space during its time evolution. As the Bloch states are defined in the reciprocal k-space, the Berry phase describes the phase picked up in the Brillouin zone [18]. The Berry connection (which is gauge dependent) is the vector potential for the Berry phase and is given by:

A

^s_β

(k) = −ihu

^s

(k)|∂

k_β

|u

^s

(k)i, (2.31) where k

_β

is one of the directions in k-space and |u

^s

(k)i are the Bloch functions.

The Berry curvature (which is gauge independent) is given by:

F

_αβ^s

(k) = ∂

_k_α

A

^s_β

(k) − ∂

_k_β

A

^s_α

(k). (2.32) The Chern number is then an intrinsic property of the band structure and is given by:

C

^s

= 1 4π

²

Z

B.Z.

F

₁₂^s

(k)d

²

k, (2.33)

where C

^s

is an integer number. For the lower bands the Berry connection will be a

constant, resulting in that F

₁₂^−,1

(k) = 0 and a zero Chern number C

^−,1

= 0. Since

the other Bloch state in the lower bands consists of the same weights but rearranged

this will also result in a zero Chern number C

^−,2

= 0. For the upper band the Chern

number will, by the same argument, be zero as well: C

^+,1

= C

^+,2

= 0. For the zero

bands the Berry curvature have a non-zero value. However, the Chern number will

still be C

^0,1

= 0. For the other Bloch state in the zero bands we get the same

results, with C

^0,2

= 0. However, the Berry curvature appears with a minus sign,

F

₁₂^0,2

(k) = −F

₁₂^0,1

(k). Since the Chern number is zero for all bands these are trivial

Chern bands.

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2.4. Wannier Functions 13

2.4 Wannier Functions

The eigenfunctions to the dice lattice that have been considered so far have been Bloch states in the reciprocal k-space. These states can be expressed in real space via Wannier functions [19]. The Wannier function for a band n is constructed as:

|Rni = L

^d

(2π)

^d

Z

B. Z.

d

^d

ke

^−ik·R

|ψ

_nk

i, (2.34)

where |ψ

nk

i is a Bloch state. The spread of the Wannier function is smaller for a larger spread in k-space and vice versa. Generally, Wannier functions are not eigenstates to a band since they have contributions from all k-states. Only in a flat band, where all k-values give the same energy, can Wannier functions constructed from the Bloch states be corresponding eigenstates. The Wannier functions are centered at a home cell R = 0. However, due to the periodicity of the lattice the entire function can be shifted by the lattice translation operator. The spread of the Wannier function therefore determines the overlap between Wannier functions at different sites. Generally, Wannier functions are not expected to be compact in space. However, for a flat band localized states are expected to exist. There then exists Wannier function that extend only to a limited number of sites. Having compact Wannier functions is a helpful property that can be used to express states in real space. Procedures that generally cannot be used, that utilizes the properties of Wannier functions as eigenfunctions, are then possible in flat bands. When considering effective models for the localized states the overlap between states are the only sites where interactions can occur. If the Wannier functions only extend to a few finite number of states the model can be projected onto those states exactly, without the need of truncation.

There exists a gauge freedom in Bloch functions where a phase can be inserted while the function remains an eigenstate. However, in the Fourier transform in equation (2.34) a shift of the phase will result in a different Wannier function.

There then exists some gauge for the Bloch function that results in the minimized spread of the Wannier function.

The Wannier functions for the states in the lowest band will be given by:

|R, −, 1i =

_(2π)^N2

R

B. Z.

d

²

k e

^−ik·R

|v

1

i

|R, −, 2i =

_(2π)^N2

R

B. Z.

d

²

k e

^−ik·R

|v

2

i. (2.35)

2.4.1 Maximally Localized in Marzari-Vanderbilt sense

Wannier functions are defined up to a gauge choice. Depending on what gauge they are calculated for the spread of the function in real space will vary. The gauge of a single band can be changed by modifying the Bloch states with a phase:

| ˜ ψ

_nk

i = e

^iϕⁿ^(k)

|ψ

_nk

i. (2.36)

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For all Bloch states there exists one preferred gauge that results in the maximal localization of the Wannier state. Due to the band degeneracy in the model methods for composite bands can be applied to find the maximally localized states [20]. Bloch states within the same degenerate band can by a unitary transformation result in new Bloch states:

| ˜ ψ

nk

i = X

m

U

_nm^(k)

|ψ

mk

i. (2.37)

The smoother the gauge the more localized the Wannier states are in real space.

For the Bloch states from equations (2.25) and (2.27), the Wannier states for the upper and lower band are localized to a finite number of sites. In figure 2.4, |ψ(r)|

²

for the Wannier functions in the lowest band is plotted for each site in the lattice. In only three unit cells (for each state) weights have a non-zero value. As these states are strictly localized they are already in the gauge leading to maximal localization.

For the zero band the Wannier functions retrieved from equation (2.30) are strictly localized in one direction and exponentially localized in the other. The two Wan- nier functions for this band can be seen in figure 2.5. The localized states can also be found using the recursion method from [16]. The localized states for the upper and lower bands are found for the weights that result in a destructive interference outside the state when a sixfold coordinated site is at the center. Strictly localized states can in the same way be found for the zero bands by setting the threefold coordinated site as the center of the states. The states in the zero band will however not be orthogonal to each other.

2.4.2 Localized states

For the lowest band two strictly localized states can be chosen. These are ”flower”- shaped states which come in two types and are localized to seven sites each. The largest weight is centered at the sixfold coordinated sites of a unit cell and each is surrounded by six weights at the surrounding neighbouring sites. These states can be seen in figure 2.6 where the distribution of weight is given by the Wannier function (as seen in figure 2.4). For the upper band the localized state are given by a change of the sign for the middle weights. The upper band is denoted by (+) and the lower band by (−). The eigenstates for the zero band will however have a different form. The localized states for the upper and lower band are explicitly given in the real space operators as:

|∗

^±₁

i = 1

√ 12 h ± √

6ˆ b

^†_j,1

+ ˆ b

^†_j,2

+ ˆ b

^†_j,3

+ ˆ b

^†_j,5

− ˆb

^†_j+e

x,2

+ ˆ b

^†_j+e

x,3

+ ˆ b

^†_j−e

y,6

i |0i (2.38)

|∗

^±₂

i = 1

√ 12

h ± √

6ˆ b

^†_j,4

+ ˆ b

^†_j,5

− ˆb

^†_j,6

− ˆb

^†_j,3

+ ˆ b

^†_j−e

x,5

+ ˆ b

^†_j−e

x,6

+ ˆ b

^†_j+e

y,2

i |0i (2.39)

where the vectors e

x

and e

y

are the unit vectors to the next extended unit cell in

the x- or y-direction. They are here given on bosonic form for the operators ˆ b

_j,i

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2.4. Wannier Functions 15

Figure 2.4: Weights of the Wannier functions of |v

1

i and |v

2

i, (eigenstates of the

lower band), where the area of each filled circle is proportional to the weight at each

site. There exists 2 types of states, one centered at site 1 of the unit cell (green)

and the other centered at site 4 (red). The weights at the centered sites have the

value

¹₂

and the weights at the surrounding sites are

₁₂¹

. Sites where the coefficient

of the wavefunction has a negative sign are marked with a darker color.

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Figure 2.5: Weights of the Wannier functions of |z

₁

i and |z

₂

i, (eigenstates of the

zero band), where the area of each filled circle is proportional to the weight at each

site. There exists 2 types of states, one centered at site 3 of the unit cell (green)

and the other centered at site 5 (red). The weights at the centered sites have the

value 0.017. Sites where the coefficient of the wavefunction has a negative sign are

marked with a darker color.

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2.4. Wannier Functions 17

Figure 2.6: The two types of localized ”flower” states in the upper and lower bands that span the degenerate bands. The rim sites with stripes have a coefficent with a negative value.

which acts upon site i in unit cell j.

The states can be seen to be orthonormal to states centered at different sites, both

for those of the same type of state and for the other type. In figure 2.7 it can be seen

how the states can be placed to span the entire lattice. The figure also includes all

configurations in which states can overlap. States further than neighbouring states

share no sites where both have weights and the overlap is there 0. For states that

are neighbouring two sites are always shared. The overlap will however be equal in

size but with opposite sign resulting in an overlap of 0. Further, for the threefold

coordinated sites a maximum of three states can overlap at once.

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Figure 2.7: Orthogonal localized states in the lower band. The states form a triangular effective lattice.

2.5 Conclusion

A dice lattice with a magnetic flux has been considered and the eigenstates of

the tight binding Hamiltonian have been found as Bloch states. The maximally

localized Wannier functions have been found for the two eigenstates that are an

orthogonal basis for the lowest band. These are two types of strictly localized states

in real space that have non-zero weights only at seven sites each. Moreover, since

there is no hopping taking place outside of these states there is also no transport

through the lattice. To study the interaction between particles the hopping can be

ignored for an effective model that instead projects the interaction so that it acts

between localized states. In the following chapter the exact projection, between

particles at sites in the lattice and localized states in the lowest band of the lattice,

is utilized to get an effective Hamiltonian.

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Chapter 3

Projecting the interaction

The localized states and flat bands described in the previous chapter are single particle properties of the dice lattice. For the non-interacting case an arbitrary filling of the lattice, states are given as product states of single particle states, which will be symmetrized or antisymmetrized for the bosonic or fermionic case respectively. If an on-site interaction is added the model can no longer be solved exactly via a Fourier transform and other analytical tools must be used to describe the properties of the resulting systems [21]. Numerical solutions are also not trivial for solving the problem at arbitrary filling, but require large computational power.

In this and the following chapters the interaction problem will be approached by formulating a model giving explicitly the effective interaction for the specific states of interest. To get an effective model for the low energy states of the interacting system a projection into a part of the Hilbert space can be made. The full model is in this chapter projected onto the localized states of the lowest band. However, to assure that the interaction keeps particles in the lower band the interaction U must be lower than the bandgap to the next band ∆: U ∆. This condition assures that the lowest states remain in the lowest single particle energy band. Due to the strictly localized states of the lower band a mapping onto these will be exact under these conditions.

The model that we consider is a Hubbard model ˆ H = ˆ H

kin

+ ˆ H

int

, where the interaction is an on-site density interaction. The on-site Hubbard interaction can then be studied by considering what the effect of the interaction is on the localized states. That is to say, the effective model is derived via a projection onto the lowest band. The full fermionic operator can be considered to be a combination of operators that act on a single band:

ˆ

c

i,σ

= X

j

[ ˆ W

⁻

(i − j)]

^∗

ˆ v

j,σ

+ X

j

[ ˆ W

⁰

(i − j)]

^∗

z ˆ

j,σ

+ X

j

[ ˆ W

⁺

(i − j)]

^∗

u ˆ

j,σ

, (3.1)

19

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where ˆ v

j,σ

is an operator for the site j in a state in the lower bands, ˆ z

j,σ

is an operator for states in the zero bands, and ˆ u

j,σ

is an operator for states in the upper bands. The coefficients ˆ W

^s

(i − j), for a band s, are those given by the Wannier function of a state centered at site i in that band. An operator ¯ c

i,σ

that acts on an entire localized state in the lower band can then be defined by projecting the operators at single sites via the Wannier function for the states. The projection acts as:

¯

c

i,σ

= X

j

[ ˆ W

⁻

(i − j)]

^∗

ˆ v

j,σ

¯ c

^†_i,σ

= X

j

W ˆ

⁻

(i − j)ˆ v

_j,σ^†

. (3.2)

The projection that instead goes to the single site operator from the flat band operators, is then given by:

ˆ

v

_j,σ

= X

i

W ˆ

⁻

(i − j)¯ c

_i,σ

ˆ v

^†_j,σ

= X

i

h ˆ W

⁻

(i − j) i

^∗

¯

c

^†_i,σ

. (3.3)

The projection onto the lower band is to project the model onto only a part of the Hilbert space. The projection can similarly be done for the rest of the Hilbert space, the two other degenerate bands. The resulting complementary operator ˜ c

_i,σ

then gives the full model together with the flat band operator. From equation (3.1):

ˆ

c

i,σ

= ¯ c

i,σ

+ ˜ c

i,σ

, (3.4) where the projection onto the rest of the Hilbert space is given by:

˜

c

i,σ

= X

j

[ ˆ W

⁰

(i − j)]

^∗

z ˆ

j,σ

+ X

j

[ ˆ W

⁺

(i − j)]

^∗

u ˆ

j,σ

. (3.5)

The projections for the fermionic and the bosonic cases will differ due to both that the Hamiltonians will be different and that the statistics of the operators are different.

3.1 Effective lattice

The new effective lattice that the effective Hamiltonian acts on is a triangular

lattice, see figure 3.1. There are two types of non-equivalent sites due to the two

types of Wannier functions spanning the degenerate lowest band. The sites of

the triangular lattice are the centered sites of each localized ”flower” state. The

matrix elements σ

ⁱⁱ_jk

= ±1 appear for the bond jk, on the side with the triangle

containing i, as a double line to indicate a negative sign. This matrix element only

affects the processes on the triangles in the effective Hamiltonian given after the

projection, equation (3.13) (fermions) or equation (3.9) (bosons). The other types

of interaction terms remain unaffected by the signs of the bonds and instead act

on an ordinary triangular lattice. The unit vectors of the triangular lattice are the

same as the unit vectors for the original dice lattice (see equation (2.8)).

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3.1. Effective lattice 21

Figure 3.1: The effective triangular lattice of the lower band when the interaction

has been projected onto the two types of localized ”flower”-states. The double

lines mean that the intetaction sign for that triangle has a negative sign σ

ⁱⁱ_jk

= −1,

when the triangle consists of the corners i, j, k. These signs are only valid for the

interactions concerning three sites. The sign of the interaction only depends on the

double lines inside of the triangle. The double line are then only valid for one side

of the bond.

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3.2 Bosons

In the Bose-Hubbard model the on-site interaction takes the form:

H ˆ

int

= U 2

X

i

ˆ

n

i

(ˆ n

i

− 1) = U 2

X

i

ˆ b

^†_i

ˆ b

^†_i

ˆ b

i

ˆ b

i

, (3.6)

where an interaction U/2 arises when more than one boson occupies a site i. If there are no restrictions to how many bosons that can occupy the same site the energy from having two bosons at the same site is U , having three bosons is 3U , and so on. The on-site interaction can be projected onto the flat band by using equation (3.3) to write the bosonic operators for the full model ˆ b

i,σ

in the operators for the localized states in the lower band ¯ b

i,σ

. If these operators are inserted into the interaction, coefficients for each projected term are given by:

V

ijkl

= U

∆

X

µ∈∆

[ ˆ W

⁻

(i − µ)]

^∗

[ ˆ W

⁻

(j − µ)]

^∗

W ˆ

⁻

(k − µ) ˆ W

⁻

(l − µ) + U

_∗

X

n∈∗

[ ˆ W

⁻

(i − n)]

^∗

[ ˆ W

⁻

(j − n)]

^∗

W ˆ

⁻

(k − n) ˆ W

⁻

(l − n), (3.7)

where Greek indices indicates the threefold connected sites and Latin indices indicates sixfold connected sites. With these indices the projected Hamiltonian is given as all combination of overlaps by summing over all possible combination of sites, for the projected operators:

H ¯

_int

= 1 2

X

i,j,k,l

V

_ijkl

¯ b

^†_i

¯ b

^†_j

¯ b

_k

¯ b

_l

. (3.8)

It can be seen directly from the localized ”flower” states that most of these terms will be 0. The only non-zero overlap between localized states (in the lower band) occurs with only one, two, or three neighbouring states at once. In the real gauge all weights are real so that [ ˆ W

⁻

(j − µ)]

^∗

= [ ˆ W

⁻

(j − µ)] and ˆ W

_↓⁻

(j − µ) = ˆ W

_↑⁻

(j − µ).

The overlap of states can be non-zero only at µ- or n-sites where a maximum of three localized states are centered at neighbouring n-sites. The indices i, j, k, l can be combined to get all combinations of contributing states in the projection.

The possible combinations will be those where all indices are the same, where three

are the same, and where two are the same. The calculation for the contributions

H ¯

_ijkl

can be found in Appendix A. As they are combined the effective Hamiltonian

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3.2. Bosons 23 becomes:

H ¯

_int

= γ

₁

X

i

¯

n

_i

(¯ n

_i

− 1) +4γ

₂

X

hi,ji

¯ n

_i

n ¯

_j

+γ

₂

X

hi,ji

h¯b

^†2_i

¯ b

²_j

+ h.c. i

+4γ

₃

X

∆(i,j,k)

σ

ⁱⁱ_jk

h

¯

n

_i

¯ b

^†_j

¯ b

_k

+ h.c. i +2γ

3

X

∆(i,j,k)

σ

ⁱⁱ_jk

h¯b

^†2_i

¯ b

j

¯ b

k

+ h.c. i ,

(3.9)

where γ

1

= U

∗

8 + U

∆

48 , γ

2

= U

∆

144 and γ

3

= U

∆

288 . Here hi, ji is a sum over all pairs of nearest neighbours. The effective lattice on which these interactions act is the triangular lattice in figure 3.1. The sum ∆(i, j, k) sums over all triangles in the effective lattice containing sites i, j, and k, where one site has two operators centered on it and the order of the other two sites has one operator centered on each. So for three sites that form a triangle there are six possible triangular terms to take into account. The signs of the matrices for the triangular sites are given by the phases on the connected links in the original lattice:

σ

_jkⁱⁱ

= exp [i(A

_iµ

+ A

_iµ

+ A

_jµ

+ A

_kµ

)]. (3.10) This matrix element has the value σ

ⁱⁱ_jk

= ±1. The four different types of triangles that result in these elements can be seen in figure 3.2. The new interactions in this effective Hamiltonian consist of a on-site density interaction, a nearest neighbour density interaction, and a pair-hopping term. There are also two terms that act on triangles which are an assisted hopping term, where density at one site can make a particle hop between the other sites of the triangle, and a pair breaking/creating term.

An effective bosonic model for the dice lattice was derived in the 2012 paper by G. M¨ oller and N.R. Cooper [7]. Due to the combinatorial factors of the operator indices considered here the resulting effective Hamiltonian does however not result in the same interaction constants as in that paper. The constants γ

1

, γ

2

, and γ

3

are given as those used for the same terms in that paper and here in equation (3.9)

three of the terms have an additional factor in front of them after the derivation

here.

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Figure 3.2: The four different configurations of three localized states in the lower

band, where a (blue) triangle has been drawn between the centering sites of the

states. The overlap between the states will depend on the weights of the rim sites

in the triangle, and these are different for the two types of localized states (red and

green). The sign of the total overlap σ

ⁱⁱ_jk

, resulting in the sign of the interaction,

will then depend on which corner i of the triangle considred. Two operators will

be centered on one of the overlapping states, the corner site i, while the other

two states have one operator centered on them. These four different triangles are

combined to form the effective lattice in figure 3.1.

An Effective Model for the Flat Bands of a Dice Lattice

Master Thesis